Class 7 Maths Chapter 3 – "A Peek Beyond the Point" (Decimals) – Question & Answer Guide: This post covers complete, page-by-page solutions for AP SCERT Class 7 Mathematics Chapter 3 – A Peek Beyond the Point (pages 135–204), the chapter that introduces decimal numbers, tenths, hundredths, thousandths, decimal place value, decimal notation, comparing and ordering decimals, addition and subtraction of decimals, and unit conversions (mm–cm–m, g–kg, rupees–paise). Perfect for AP Class 7 students, teachers and parents searching for APSCERT Class 7 Maths Chapter 3 solutions, decimal numbers worksheet answers, Figure it Out 3.1 to 3.6 answers, and Try This answer key. All exercises are solved step-by-step in the exact order in the textbook.
📘 Unit 3: A Peek Beyond the Point – Full Solutions
Chapter 3 – "A Peek Beyond the Point" teaches how a unit can be split into 10 equal tenths, each tenth into 10 equal hundredths, and further into thousandths, extending the Indian place value system beyond whole numbers using the decimal point. Students learn to read, write, compare and order decimal numbers, locate them on a number line, add and subtract decimals, and convert between related units of length (mm, cm, m), weight (g, kg) and money (paise, rupees).
📍 Pages 136–138: Introduction – Measuring Screws
Discussion Question: Which scale helped you measure the length of the screws accurately? Why?
Answer: The ruler that was divided into 10 equal smaller parts (tenths) between each cm mark helped measure the screws accurately, because it allowed us to record lengths that fall between two whole centimetres, like 2 7/10 cm.
Discussion Question: What is the meaning of 2 7/10 cm (the length of the first screw)?
Answer: It means the screw's length is 2 whole centimetres plus 7 out of 10 equal parts of the next centimetre, i.e. we go from 0 to 2 and then take 7 parts of 1/10 more.
| Length | Can be read as |
|---|---|
| 4 3/10 cm | four and three-tenth centimeters |
| 5 8/10 cm | five and eight-tenth centimeters |
| 8 4/10 cm | eight and four-tenth centimeters |
Measuring the screws (right column) & objects: Sample readings (students should measure their own printed copy for the exact value): the three right-side screws measure approximately 1 8/10 cm, 2 4/10 cm and 2 9/10 cm. For the eraser, pencil and crayon on the next page, sample readings are: eraser ≈ 2 3/10 units, pencil ≈ 4 1/10 units, crayon ≈ 1 5/10 units.
📍 Pages 140–148: Section 3.1 – A Tenth Part
Try This (Page 142): Arrange the following lengths in increasing order:
(a) 9/10 (b) 1 7/10 (c) 130/10 (d) 13 1/10 (e) 10 5/10 (f) 7 6/10 (g) 6 7/10 (h) 4/10
Answer: Converting to decimals: (a)=0.9, (b)=1.7, (c)=13, (d)=13.1, (e)=10.5, (f)=7.6, (g)=6.7, (h)=0.4
Increasing order: (h) 4/10 < (a) 9/10 < (b) 1 7/10 < (g) 6 7/10 < (f) 7 6/10 < (e) 10 5/10 < (c) 130/10 < (d) 13 1/10
Try This (Page 142): Arrange in increasing order: 4 1/10, 4/10, 41/10, 41 1/10
Answer: 4 1/10 = 4.1, 4/10 = 0.4, 41/10 = 4.1, 41 1/10 = 41.1
Increasing order: 4/10 (0.4) < 4 1/10 = 41/10 (both 4.1) < 41 1/10 (41.1)
✏️ Figure it Out – 3.1 (Page 148)
1. A Celestial Pearl Danio's length is 2 4/10 cm, and the length of a Philippine Goby is 9/10 cm. What is the difference in their lengths?
Answer: 2 4/10 cm = 2.4 cm and 9/10 cm = 0.9 cm.
Difference = 2.4 − 0.9 = 1.5 cm
2. How big are these fish compared to your finger?
Answer: Both fish are much smaller than a finger — a finger is usually 6–8 cm long, while these fish are only about 1–2.5 cm long, so they are roughly 3 to 4 times shorter than a finger.
3. Observe the given sequences of numbers. Identify the change after each term and extend the pattern:
(b) 8 2/10, 8 7/10, 9 2/10, 9 7/10, 10 2/10, 10 7/10, 11 2/10 (add 5/10)
(c) 7 6/10, 8 7/10, 9 8/10, 10 9/10, 12, 13 1/10 (add 1 1/10)
(d) 5 7/10, 5 3/10, 4 9/10, 4 5/10, 4 1/10, 3 7/10 (subtract 4/10)
(e) 13 5/10, 13, 12 5/10, 12, 11 5/10, 11, 10 5/10 (subtract 5/10)
(f) 11 5/10, 10 4/10, 9 3/10, 8 2/10, 7 1/10, 6, 4 9/10 (subtract 1 1/10)
📍 Pages 150–160: Section 3.2 – A Hundredth Part
Discussion Questions (Page 150):
What is the length of the smaller part? How many such smaller parts make a unit length?
Answer: The length of the smaller part is 1/100 of a unit; 100 such parts make one unit.
How many one-hundredths make one-tenth? Can we say the length is 4 units and 45 one-hundredths?
Answer: 10 one-hundredths make one-tenth. Yes — 4 4/10 5/100 = 4 + 40/100 + 5/100 = 4 45/100, so the length can indeed be called 4 units and 45 one-hundredths.
✏️ Figure it Out – 3.2 (Pages 152–154)
1. Observe the figure. Fill the lengths in the empty boxes on the ruler (marks between 0 and 2 divided into hundredths).
Answer: Following the pattern shown (each small division = 1/100), the empty boxes are filled as: 40/100, 1 (=100/100), 160/100, 180/100, 190/100, 2 (=200/100) — each box equals the tenth mark expressed in hundredths.
2. For the lengths shown, write the measurements and read the measures in words:
b) 14.9 cm — read as fourteen and nine-tenth centimeters
c) 7.5 cm — read as seven and five-tenth centimeters
d) 9.8 cm — read as nine and eight-tenth centimeters
(Exact values should be verified against your own textbook diagram; the method is: count the whole units then the extra tenth divisions.)
3. In each group, identify the longest and shortest lengths:
| Group | Values (as decimals) | Longest | Shortest |
|---|---|---|---|
| (a) 3/10, 3/100, 33/100 | 0.3, 0.03, 0.33 | 33/100 (0.33) | 3/100 (0.03) |
| (b) 3 1/10, 30/10, 1 3/10 | 3.1, 3.0, 1.3 | 3 1/10 (3.1) | 1 3/10 (1.3) |
| (c) 45/100, 54/100, 5/10, 4/10 | 0.45, 0.54, 0.5, 0.4 | 54/100 (0.54) | 4/10 (0.4) |
| (d) 3 6/10, 3 6/100, 3 6/10 6/100 | 3.6, 3.06, 3.66 | 3 6/10 6/100 (3.66) | 3 6/100 (3.06) |
| (e) 8/10, 2/100, 9/100, 1 8/100 | 0.8, 0.02, 0.09, 1.08 | 1 8/100 (1.08) | 2/100 (0.02) |
| (f) 7 3/10 5/100, 7 5/10, 7 41/100 | 7.35, 7.5, 7.41 | 7 5/10 (7.5) | 7 3/10 5/100 (7.35) |
| (g) 65/10, 15/100, 5 87/100, 5 7/100 | 6.5, 0.15, 5.87, 5.07 | 65/10 (6.5) | 15/100 (0.15) |
Solve by converting to hundredths (Page 158): What is the sum of 15 3/10 4/100 and 2 6/10 8/100?
Answer: 15.34 + 2.68 = 18.02
✏️ Figure it Out – 3.3 (Pages 158–160)
Find the sums and differences:
(b) 9 5/10 7/100 + 2 1/10 3/100 = 9.57 + 2.13 = 11.70
(c) 15 6/10 4/100 + 14 3/10 6/100 = 15.64 + 14.36 = 30.00 (30)
(d) 7 7/100 − 4 4/100 = 7.07 − 4.04 = 3.03
(e) 8 6/100 − 5 3/100 = 8.06 − 5.03 = 3.03
(f) 12 6/10 2/100 − 9 9/10 9/100 = 12.62 − 9.99 = 2.63
📍 Pages 160–172: Section 3.3 – Decimal Place Value
Try This (Page 170): Make a place value table. Write each quantity in decimal form and read the number:
(b) 1 ten and 5 tenths → 10.5 — read as ten point five
(c) 4 ones and 6 hundredths → 4.06 — read as four point zero six
(d) 1 hundred, 1 one and 1 hundredth → 101.01 — read as one hundred one point zero one
(e) 8/100 and 9/10 → 0.98 — read as zero point nine eight
(f) 5/100 → 0.05 — read as zero point zero five
(g) 1/10 → 0.1 — read as zero point one
(h) 2 1/100, 4 1/10 and 7 7/1000 → 2.01, 4.1, 7.007
Try This (Page 172): Write these quantities in decimal form:
(b) 105 tenths → 10.5
Discussion: How many thousandths make one unit / one tenth / one hundredth? How many tenths make one ten? How many hundredths make one ten?
(d) 100 tenths make 1 ten (e) 1000 hundredths make 1 ten
📍 Pages 172–182: Section 3.4 – Units of Measurement
Try This (Page 172): Fill in the blanks (mm ↔ cm):
| mm | cm |
|---|---|
| 12 mm | 1.2 cm (given) |
| 56 mm | 5.6 cm (given) |
| 70 mm | 7.0 cm |
| 9 mm | 0.9 cm |
| 134 mm | 13.4 cm |
| 2036 mm | 203.6 cm |
Try This (Page 176): Fill in the blanks (cm ↔ m):
| cm | m |
|---|---|
| 36 cm | 0.36 m |
| 50 cm | 0.50 m |
| 89 cm | 0.89 m |
| 4 cm | 0.04 m |
| 325 cm | 3.25 m |
| 207 cm | 2.07 m |
Try This (Page 178): Fill in the blanks (g ↔ kg):
| g | kg |
|---|---|
| 465 g | 0.465 kg |
| 68 g | 0.068 kg |
| 1560 g | 1.560 kg |
| 704 g | 0.704 kg |
| 560 g | 0.56 kg |
| 2500 g | 2.5 kg |
Try This (Page 180): Fill in the blanks (rupee ↔ paise):
| paise | rupee |
|---|---|
| 10 p | ₹0.10 |
| 5 p | ₹0.05 |
| 36 p | ₹0.36 |
| 50 p | ₹0.50 |
| 99 p | ₹0.99 |
| 250 p | ₹2.50 |
📍 Pages 182–190: Section 3.5 – Locating and Comparing Decimals
Discussion: Name all divisions between 1 and 1.1 on the number line.
Answer: 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09 (each division = 0.01)
✏️ Figure it Out – 3.4 (Pages 184–190)
1. Which of these are the same: 4.5, 4.05, 0.405, 4.050, 4.50, 4.005, 04.50?
Answer: 4.5 = 4.50 = 04.50 (all equal 4.5, since adding zeros after the last decimal digit, or before the whole number, doesn't change value). Also 4.05 = 4.050 (both equal 4.05). The numbers 0.405 and 4.005 are different from all the others and from each other.
2. Identify the decimal number in the last number line in Figure (b) denoted by '?'.
Answer: Using the same zooming method as Figure (a) — zoom into the segment 0–10, then a unit, then a tenth, then a hundredth — the marked point corresponds to a number with three decimal places, read off directly from the final magnified line (for example, a point like 8.375, following the same step-by-step zooming shown for 4.185 in Figure (a)).
3. Make such number lines for the decimal numbers: (a) 9.876 (b) 0.407
Answer: (a) 9.876: Zoom 1: mark between 9 and 10; Zoom 2: divide into tenths and mark between 9.8 and 9.9; Zoom 3: divide that into hundredths and mark between 9.87 and 9.88; Zoom 4: divide into thousandths and mark at 9.876.
(b) 0.407: Zoom 1: mark between 0 and 1; Zoom 2: divide into tenths and mark between 0.4 and 0.5; Zoom 3: divide into hundredths and mark between 0.40 and 0.41; Zoom 4: divide into thousandths and mark at 0.407.
4. a) In the number line (5 to 10), what decimal numbers do boxes 'a', 'b', 'c' denote? (b = 7.5 is given)
Answer: Since there are 5 units between 5 and 10 divided into 10 equal parts, each division = 1/2 unit = 0.5. So, moving along the line: a = 6, b = 7.5 (given), c = 9 (values depend on the exact position marked in the textbook figure; the method is to count divisions of 0.5 from 5).
4. b) Using similar reasoning, find the decimal numbers in the boxes below (d, e between 8 and 8.1; f, g, h between 4.3 and 4.8).
Answer: Between 8 and 8.1, each small division = 0.01, so d = 8.03, e = 8.07 (approx., read from position). Between 4.3 and 4.8, each division = 0.05, so f = 4.4, g = 4.55, h = 4.7 (approx., read from position).
Which is larger: 6.456 or 6.465?
Answer: Both have 6 units and 4 tenths. Comparing hundredths: 6.456 has 5 hundredths, 6.465 has 6 hundredths. So 6.465 is larger.
Which decimal number is greater?
(b) 3.81 or 13.800 → 13.800 is greater
(c) 1.009 or 1.090 → 1.090 is greater
Closest Decimals:
Which among 3.56, 3.65, 3.099 is closest to 4? → 3.65 (difference = 0.35, the smallest)
Which among 0.8, 0.69, 1.08 is closest to 1? → 1.08 (difference = 0.08, the smallest)
Math Talk: Using digits 4, 1, 8, 2, 5 exactly once, make a decimal number as close as possible to 25.
Answer: Using the format XX.XXX, the closest number to 25 is 25.148 (distance from 25 = 0.148), formed by placing the two largest suitable digits (2, 5) as the whole part and arranging the remaining digits (1, 4, 8) in ascending order as the decimal part to keep the number just slightly above 25.
📍 Pages 190–198: Section 3.6 – Addition and Subtraction of Decimals
Try This (Page 194): Write the detailed place value computation for 84.691 − 77.345, and its compact form.
Answer: 84.691 − 77.345 = 7.346 (In compact column form: units 4−7 needs borrowing, so regroup: 84.691 − 77.345 = 7.346)
✏️ Figure it Out – 3.5 (Page 194)
1. Find the sums
(e) 29.19 + 9.91 = 39.10 (f) 0.934 + 0.6 = 1.534 (g) 0.75 + 0.03 = 0.78 (h) 6.236 + 0.487 = 6.723
2. Find the differences
(e) 17 − 0.05 = 16.95 (f) 34.505 − 18.1 = 16.405 (g) 9.9 − 9.09 = 0.81 (h) 6.236 − 0.487 = 5.749
Decimal Sequences: 4.4, 4.8, 5.2, 5.6, 6.0, ... Continue and write the next 3 terms.
Answer: Adding 0.4 each time: 6.4, 6.8, 7.2
Identify the change and write the next 3 terms for each sequence:
(b) 25.75, 26.25, 26.75, ... → 27.25, 27.75, 28.25 (add 0.5)
(c) 10.56, 10.67, 10.78, ... → 10.89, 11.00, 11.11 (add 0.11)
(d) 13.5, 16, 18.5, ... → 21, 23.5, 26 (add 2.5)
(e) 8.5, 9.4, 10.3, ... → 11.2, 12.1, 13.0 (add 0.9)
(f) 5, 4.95, 4.90, ... → 4.85, 4.80, 4.75 (subtract 0.05)
(g) 12.45, 11.95, 11.45, ... → 10.95, 10.45, 9.95 (subtract 0.5)
(h) 36.5, 33, 29.5, ... → 26, 22.5, 19 (subtract 3.5)
Estimating Sums and Differences (Discussion): Is Karthik's claim true — that the sum of two decimals is always greater than the sum of their whole-number parts, and less than 2 more than that sum?
Answer: Yes, this is true for any two positive decimal numbers, because each decimal part is between 0 and 1 (but less than 1), so the sum of the two decimal parts is between 0 and 2. For example, 25.936 + 8.202: whole parts sum to 25+8=33; actual sum = 34.138, which is greater than 33 and less than 33+2=35. The same logic works for 25.93603259 + 8.202 as well — the actual sum still lies strictly between 33 and 35.
📍 Pages 196–204: Section 3.7 – More on the Decimal System
Discussion: How could the Amsterdam City Council's error and the Air Canada fuel error be avoided?
Answer: Both errors happened because of confusing units (euro-cents vs euros; pounds vs kilograms). They could have been avoided by always double-checking the unit being used before entering or reading a decimal value, and by having a second person verify important calculations before payments or fuel loading were finalised.
Discussion: If the bus reaches the station "4.5 hours post noon", when will it arrive?
Answer: 0.5 hours = 5 parts out of 10 equal parts of an hour = 5 × 6 minutes = 30 minutes. So the bus reaches the station at 4:30 p.m. (not 4:05, 4:50 or 4:25 p.m.)
✏️ Figure it Out – 3.6 (Pages 200–204)
1. Convert the following fractions into decimals:
2. Convert the following decimals into a sum of tenths, hundredths and thousandths:
(b) 1.02 = 1 + 0/10 + 2/100
(c) 0.8 = 8/10
(d) 0.362 = 3/10 + 6/100 + 2/1000
3. What decimal number does each letter represent (number line 6.4 to 6.6, marks a, c, b)?
Answer: Each small division on this line = 0.01. Based on their positions: a = 6.45, c = 6.50, b = 6.52 (exact readings should be confirmed against your printed number line).
4. Arrange the following quantities in descending order:
(b) 2.567, 2.675, 2.768, 2.499, 2.698 → 2.768 > 2.698 > 2.675 > 2.567 > 2.499
(c) 4.678 g, 4.595 g, 4.600 g, 4.656 g, 4.666 g → 4.678 > 4.666 > 4.656 > 4.600 > 4.595
(d) 33.13 m, 33.31 m, 33.133 m, 33.331 m, 33.313 m → 33.331 > 33.313 > 33.31 > 33.133 > 33.13
5. Using the digits 1, 4, 0, 8, and 6, make:
(b) the smallest possible decimal number between 100 and 1000 → 104.68 (smallest 3-digit whole number is 104, and remaining digits 6, 8 form the smallest decimal part .68)
6. Will a decimal number with more digits be greater than a decimal number with fewer digits?
Answer: No. The number of digits does not decide the value — the place value of each digit matters. For example, 0.9 (1 digit) is greater than 0.12345 (5 digits), because 0.9 has a much bigger digit in the tenths place.
7. Pratap purchases 0.25 kg beans, 0.3 kg carrots, 0.5 kg potatoes, 0.2 kg capsicums and 0.05 kg ginger. Calculate the total weight.
Answer: 0.25 + 0.3 + 0.5 + 0.2 + 0.05 = 1.3 kg
8. Rajesh supplies 3.79 L, 4.2 L and 4.25 L of milk in the first three days. In 6 days he supplies 25 litres in total. Find the quantity supplied in the last three days.
Answer: First 3 days total = 3.79 + 4.2 + 4.25 = 12.24 L
Last 3 days = 25 − 12.24 = 12.76 L
9. Sudhir weighed 35.75 kg in January and 34.50 kg in February. Has he gained or lost weight? How much is the change?
Answer: 35.75 − 34.50 = 1.25 kg. Since his weight decreased, Sudhir has lost weight by 1.25 kg.
10. Extend the pattern: 5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17, ___, ___
Answer: The pattern alternates between adding about 0.9/0.89 and subtracting 0.01 (with an occasional larger drop). Continuing it: 7.07, 7.06
11. How many millimeters make 1 kilometer?
Answer: 1 km = 1000 m, and 1 m = 1000 mm, so 1 km = 1000 × 1000 = 10,00,000 mm (one million mm)
12. Indian Railways charges 45 paise per passenger for optional travel insurance. If 1 lakh people opt for insurance in a day, what is the total insurance fee paid?
Answer: Total = 45 paise × 1,00,000 = 45,00,000 paise = 45,00,000 ÷ 100 = ₹45,000
13. Which is greater?
(b) One-hundredth or 90 thousandths → 0.01 or 0.090 → 90 thousandths is greater
(c) One-thousandth or 90 hundredths → 0.001 or 0.90 → 90 hundredths is greater
14. Write the decimal forms of the quantities mentioned:
(b) 12 tens and 12 tenths = 120 + 1.2 = 121.2
(c) 10 tens, 10 ones, 10 tenths and 10 hundredths = 100 + 10 + 1.0 + 0.10 = 111.1
(d) 25 tens, 25 ones, 25 tenths and 25 hundredths = 250 + 25 + 2.5 + 0.25 = 277.75
15. Using each digit 0–9 not more than once, fill the boxes (format: [ ].[ ][ ][ ] + [ ].[ ][ ][ ]) so the sum is closest to 10.5.
Answer: One very close solution is: 6.158 + 4.297 = 10.455 (distance from 10.5 = only 0.045). This uses the 8 distinct digits 6,1,5,8,4,2,9,7. (This is an open exploration question — other combinations of digits that give a sum near 10.5 are also acceptable.)
16. Write the following fractions in decimal form:
✅ Chapter Summary
Chapter 3 – "A Peek Beyond the Point" teaches how a unit can be split into 10 equal tenths, each tenth into 10 equal hundredths, and further into thousandths, extending the Indian place value system beyond whole numbers using the decimal point. Students learn to read, write, compare and order decimal numbers, locate them on a number line, add and subtract decimals, and convert between related units of length (mm, cm, m), weight (g, kg) and money (paise, rupees).
